How can I thicken mesh hair?
Since you are modeling the hair and going for an anime style. The Solidify modifier should do this quite well. It might seem subtle but you will get the depth you want.

Flat mesh hair.

Mesh hair with solidify modifier.
After using the modifier, you can apply it and tweak the mesh even further to get better results. Using the modifier is better than just extruding and tweaking as it will do this evenly for your entire mesh.
You can get some unique styles as well if you smooth the hair with a subsurf modifier and play with the settings.